Theoretical Studies of Proton-Coupled Electron Transfer Reactions via the Mixed Quantum-Classical Liouville Approach

Transcription

1 Theoretical Studies of Proton-Coupled Electron Transfer Reactions via the Mixed Quantum-Classical Liouville Approach by Farnaz A. Shakib A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Chemistry University of Alberta Farnaz Alipour Shakib, 2016

2 Abstract The nonadiabatic dynamics of model proton-coupled electron transfer (PCET) reactions were investigated for the first time using a surface-hopping algorithm based on the solution of the mixed quantum-classical Liouville (MQCL) equation. This method provides a rigorous treatment of quantum decoherence effects in the dynamics of mixed quantum-classical systems, which is lacking in Tully s fewest switches surface hopping (FSSH) approach commonly used for simulating PCET reactions. Within the MQCL approach, the transferring proton and electron are treated quantum mechanically while the remainder of the system, including the donor, acceptor, and solvent, is treated classically. The classical degrees of freedom (DOF) are evolved on both single adiabatic potential energy surfaces (PESs) and on coherently coupled pairs of adiabatic PESs, while in FSSH they are evolved only on single adiabatic PESs. To demonstrate the applicability of MQCL approach to the study of PCET, we focused on a reduced model that had been previously studied using FSSH. This model is comprised of three DOF, including a proton, an electron, and a collective solvent coordinate. The proton and electron are allowed to transfer in one dimension between two point charges, representing the donor and the acceptor, which are positioned at a fixed distance from each other. Both concerted and sequential [either proton transfer before electron transfer (i.e., PT-ET) or electron transfer before proton transfer (i.e., ET-PT)] PCET reactions were studied within the context of this model. We studied these mechanisms in detail under various subsystem-bath coupling conditions and gained insights into the dynamical principles underlying these reactions. Notably, an examination of the trajectories which successfully undergo PCET (i.e., both the proton and electron, initially near the donor, transfer to the acceptor) reveals that the system spends the majority of its time on the mean of two coherently coupled adiabatic PESs. While on this mean surface, the classical DOF evolve differently than ii

3 on the other surfaces and an observable of interests acquires a phase factor. Fluctuations of the classical coordinates can cause this phase factor to oscillate in time differently for each trajectory and, as a result, averaging over an ensemble of trajectories can lead to destructive interference when calculating an expectation value of an observable. In this way, the MQCL approach is able to incorporate decoherence, which is not captured in the FSSH approach. Due to this fundamental difference between the two methods, the percentages of successful PCET reactions obtained via MQCL drastically differ from those obtained via FSSH. To gain insight into the differences between the MQCL and FSSH approaches for calculating PCET observables, we calculated the time-dependent populations of the ground, first-excited, and second-excited adiabatic states for the ET-PT mechanism in the same PCET model and compared them to both the exact quantum and FSSH results. We found that the MQCL population profiles show a significant improvement over the FSSH ones, and are in good agreement with the exact quantum results. All of the mean PESs were shown to play a direct role in the dynamics of the state populations. Interestingly, our results showed that the population transfer to the second-excited state can be mediated by the dynamics on the mean of the ground and second-excited state PESs, via a sequence of nonadiabatic transitions that bypasses the first-excited state PES altogether. This is made possible by nonadiabatic transitions between different mean surfaces, which is the manifestation of coherence transfer in MQCL dynamics. We also investigated the effect of the strength of the coupling between the proton/electron and the collective solvent coordinate on the state population dynamics. Drastic changes in the population dynamics are observed, which can be understood in terms of the changes in the PESs and the nonadiabatic couplings. In addition, we investigated the state population dynamics in the PT-ET and concerted versions of the model. The PT-ET results revealed the participation of all of the mean surfaces, albeit in different proportions iii

4 compared to the ET-PT reaction, while the concerted results revealed that the mean of the ground and first-excited state PESs only plays a role. We finally present a derivation of a novel mixed quantum-classical formula for calculating PCET rate constants that starts from the quantum mechanical expression for the time integral of the flux-flux correlation function. The resulting time-dependent rate coefficient has a dynamical part involving MQCL time evolution of the product species operator and an equilibrium part. This formula is general, not requiring any prior assumptions about the parameters of the system. Furthermore, since MQCL dynamics is used to evolve the product species operator, this approach treats decoherence on a more solid footing than the FSSH-based methods. The applicability of this formula is demonstrated on an extended version of the previously studied PCET model, where now the collective solvent coordinate is coupled to a harmonic oscillator bath. Our result for the rate constant is found to be in good agreement with the numerically exact result obtained via the quasi-adiabatic path integral method. iv

5 Preface Chapter 2 of this thesis contains the following published article: Shakib, F. A. and Hanna, G. An analysis of model proton-coupled electron transfer reactions via the mixed quantum-classical Liouville approach, J. Chem. Phys., 2014, 141, Chapter 3 of this thesis contains the following published article: Shakib, F. A. and Hanna, G. New insights into the nonadiabatic state population dynamics of model proton-coupled electron transfer reactions from the mixed quantum-classical Liouville approach, J. Chem. Phys., 2016, 144, My contribution to these articles included aiding in the selection of an appropriate PCET model, writing computer codes for simulating the dynamics of this model and analyzing data, running the simulations, preparing the draft of the manuscript, and helping my supervisor in bringing the manuscripts to their final forms. v

6 Acknowledgements I would like to thank the Alberta Innovates Future Technology (AIFT) and Natural Sciences and Engineering Research Council of Canada (NSERC) for financial support over the years. I would also like to thank WestGrid and Compute/Calcul Canada for access to their computational resources over the years. IamgreatlyindebtedtoProf. GabrielHannaforallhistime, energy, andpatienceduring the course of my PhD. His thoughtful guidance over the years will always be appreciated. I would also like to thank the members of the Theoretical and Computational Chemistry group in our department for their friendly support and encouragement. I am also grateful to the members of my examination committee (Profs. Mariusz Klobukowski, Yunji Xu, Frederick West, and Juli Gibbs-Davis) for taking the time to read my thesis and providing thoughtful comments. Finally, I am grateful to Prof. Thomas F. Miller from the California Institute of Technology for agreeing to be the external referee of this thesis and providing constructive comments/suggestions. vi

7 To my everlasting ray of hope, Mohammad vii

8 Abbreviations CW classical Wigner DOF degree of freedom ET electron transfer ET-PT electron transfer followed by proton transfer FSSH fewest switches surface hopping GFSH global flux surface hopping HAT hydrogen-atom transfer MD molecular dynamics MQCD mixed quantum-classical dynamics MQCL mixed quantum-classical Liouville MS-EPT multi site electron-proton transfer OEC oxygen evolving complex PCET proton-coupled electron transfer PES potential energy surface PT proton transfer PT-ET proton transfer followed by electron transfer QCLE quantum-classical Liouville equation RPMD ring-polymer molecular dynamics SQUASH second-quantized surface hopping SSTP sequential short-time propagation viii

9 Contents Acknowledgements vi Abbreviations Contents List of Figures List of Tables viii ix xi xv 1 Introduction PCET in biology and chemistry Elementary steps of PCET Theoretical challenges Mixed quantum-classical dynamics (MQCD) Fewest switches surface hopping (FSSH) Mixed quantum-classical Liouville (MQCL) approach FSSH vs. MQCL Outline of thesis An Analysis of Model Proton-Coupled Electron Transfer Reactions via the Mixed Quantum-Classical Liouville Approach Introduction PCET model Simulation details Results Adiabatic potential energy surfaces and nonadiabatic couplings Nonadiabatic dynamics Concerted PCET mechanism Sequential PT-ET mechanism Sequential ET-PT mechanism Role of phase factor Effect of proton/electron-solvent coupling strength Concluding remarks ix

10 Contents 3 New Insights into the Nonadiabatic State Population Dynamics of Model Proton-Coupled Electron Transfer Reactions from the Mixed Quantum- Classical Liouville Approach Introduction MQCL dynamics of an observable Simulation details Results and discussion Sequential ET-PT mechanism Potential energy surfaces and nonadiabatic couplings Time-dependent state populations and coherences Effect of proton/electron-solvent coupling strength Sequential PT-ET mechanism Potential energy surfaces and nonadiabatic couplings Time-dependent state populations and coherences Concerted PCET mechanism Potential energy surfaces and nonadiabatic couplings Time-dependent state populations and coherences Concluding remarks Calculating Proton-Coupled Electron Transfer Rate Constants via the Mixed Quantum-Classical Liouville Approach Introduction PCET model Derivation of the time-dependent rate coefficient General quantum-classical expression PCET rate coefficient Simulating the time-dependent rate coefficient Sampling the thermal equilibrium distributions Simulating the time evolution of the product species operator Calculating the equilibrium density of the reactant species Results for PCET model Solving the time-independent Schrödinger equation Results Conclusion Conclusion Concluding remarks Future plans x

11 List of Figures 1.1 (a) Different components of Photosystem II, (b) the optimized structure of OEC, reprinted (adapted) with permission from Ref. 1. Copyright (2015) American Chemical Society The Kok cycle representing the four oxidation states of OEC, reprinted(adapted) with permission from Ref. 2. Copyright (2015) American Chemical Society An example of a concerted electron-proton transfer(ept), reprinted(adapted) with permission from Ref. 3. Copyright (2015) American Chemical Society An example of hydrogen atom transfer, reprinted (adapted) with permission from Ref. 3. Copyright (2015) American Chemical Society A schematic representation of concerted and sequential PCET reactions. D represents the donor and A represents the acceptor species Schematic representation of the one-dimensional PCET model studied in this work. The protonic and electronic potentials (V p and V e, respectively) are superimposed onto the donor(d) acceptor(a) axis. The protonic and electronic coordinates (q p and q e, respectively) are defined relative to the donor The 2D PESs as a function of q p and q e at different solvent configurations for concerted PCET (top row), sequential PT-ET (middle row), and the sequential ET-PT (bottom row) mechanisms. Q values are in Å but q p and q e values in this figure are given in a.u Adiabatic potential energy surfaces as a function of the solvent coordinate, Q, for the concerted (top), PT-ET (middle), and ET-PT (bottom) reactions. The blue curves correspond to the ground and first excited state surfaces, while the red curves correspond to the mean surfaces The ground and the first excited state wave functions for the concerted PCET as a function of the solvent configuration Potential energy surfaces of PT-ET (top), and ET-PT (bottom) mechanisms as a function of solvent configuration The ground state wave functions for the sequential PCET mechanisms as a function of the solvent configuration xi

12 List of Figures 2.7 Nonadiabatic coupling strength, d 12, as a function of the solvent coordinate, Q, computed for concerted PCET (top panels), PT-ET (middle panels), and ET-PT (bottom panels). The insets zoom in on the sections corresponding to lower d 12 values. Left panels: Lower proton/electron-solvent coupling. The coupling constants for concerted PCET, PT-ET, and ET-PT are C sp = C se = , C sp /C se = /5 10 4, and C sp /C se = /1 10 3, respectively. Right panels: Higher proton/electron-solvent coupling. The coupling constants for concerted PCET, PT-ET, and ET-PT are C sp = C se = , C sp /C se = /2 10 3, andc sp /C se = /4 10 3, respectively Percentage of trajectories which successfully undergo a PCET reaction (denoted by % PCET) and the percentage of these trajectories which contain 0, 2, 4, 6, 8, and 10 nonadiabatic transitions (denoted by 0 jumps, 2, jumps, etc.) as a function of the initial momentum for the concerted (top), PT-ET (middle), and ET-PT (bottom) reactions Percentage of time(out of the entire ensemble) spent on the ground state(1,1), mean [(1,2) and (2,1)], and first excited state (2,2) surfaces as a function of the initial momentum for the concerted (top), PT-ET (middle), and ET-PT (bottom) mechanisms Average of the real part of W(t), Re[W(t)], over an ensemble of trajectories starting from Q i =-0.24, -0.58, and -0.5 Å and initial momentum distributions (corresponding to a temperature of 300 K) centred around different P i, for the concerted (top), PT-ET (middle), and ET-PT (bottom) reactions, respectively Percentage of trajectories which successfully undergo a PCET reaction as a function of the coupling strength between the proton/electron and the solvent for various initial momenta, P i, for the concerted (top), PT-ET (middle), and ET-PT (bottom) reactions. The x-axis labels, 1 7, represent seven sets of coupling constants {C sp,c se } (values given in Table 2), listed in order of increasing strength Adiabatic potential energy surfaces for the seven different proton/electronsolvent coupling strengths (listed in order of increasing strength from 1 to 7), computed for the concerted mechanism with P i = 10.0 a.u. Note that for coupling strength 7, the reaction barrier is sufficiently high that the system remains in the donor state for this P i The percentage of trajectories which undergo adiabatic reaction as a function of the coupling strength between the proton/electron and the solvent for various initial momenta, P i, for the concerted (top), sequential PT-ET (middle), and ET-PT (bottom) reactions. The x axis labels 1 7 denote seven sets of coupling constants {C sp,c se }, in order of increasing strength. The results were obtained using the MQCL surface-hopping method Average of the real part of W(t), Re[W(t)], over an ensemble of trajectories starting from Q i =-0.24, -0.58, and -0.5 Å and an initial momentum distribution (corresponding to a temperature of 300 K) centred around P i = 10, 10, and 28 a.u. for the concerted (top), PT-ET (middle), and ET-PT (bottom) reactions, respectively, calculated for three different sets of proton/electronsolvent coupling strengths xii

13 List of Figures 3.1 Top panel: Ground (1,1), first excited (2,2), and second excited (3,3) state adiabatic PESs as a function of the solvent coordinate, Q, for the ET-PT model. The blue curves correspond to these surfaces, the red curves correspond to the mean surfaces between adjacent surfaces, and the green curve corresponds to the mean surface between the ground and second excited state surfaces. Bottom panel: Absolute values of the nonadiabatic coupling matrix elements, d 12, d 23, and d 13 as a function of the solvent coordinate, Q Top panel: Time evolutions of the populations of the ground (1,1), firstexcited (2,2), and second-excited (3,3) adiabatic states for the ET-PT model, calculated via MQCL, FSSH, and exact quantum dynamics. Middle panel: The percentage of trajectories on each surface as a function of time. Bottom panel: Time evolutions of the (1,2), (2,3), and (1,3) coherences, calculated via MQCL Left panels: Ground (1,1), first excited (2,2), and second excited (3,3) state adiabatic PESs as a function of the solvent coordinate, Q, for the ET-PT model, corresponding to different proton/electron-solvent coupling strengths: C sp = /C se = (weakest), C sp = /C se = (intermediate), and C sp = /C se = (strongest). The blue curves correspond to the (1,1), (2,2), and (3,3) surfaces, the red curves correspond to the mean surfaces between adjacent surfaces, and the green curve corresponds to the mean surface between the ground and second excited state surfaces. Right panels: Close-ups of the surfaces around Q = 0 Å Absolutevaluesofthenonadiabaticcouplingmatrixelements d 12 (toppanel), d 23 (middlepanel),and d 13 (bottompanel),asafunctionofthesolventcoordinate, Q, for the ET-PT model, corresponding to different proton/electronsolvent coupling strengths: C sp = /C se = (weakest), C sp = /C se = (intermediate), and C sp = /C se = (strongest) Time evolutions of the populations of the ground (1,1), first excited (2,2), and second excited (3,3) adiabatic states for the ET-PT model, calculated via MQCL, corresponding to different proton/electron-solvent coupling strengths: C sp = /C se = (weakest), C sp = /C se = (intermediate), and C sp = /C se = (strongest) Top panel: Ground (1,1), first excited (2,2), and second excited (3,3) state adiabatic PESs as a function of the solvent coordinate, Q, for the PT-ET model. The blue curves correspond to these surfaces, the red curves correspond to the mean surfaces between adjacent surfaces, and the green curve corresponds to the mean surface between the ground and second excited state surfaces. Bottom panel: Absolute values of the nonadiabatic coupling matrix elements, d 12, d 23, and d 13 as a function of the solvent coordinate, Q Top panel: Time evolutions of the populations of the ground (1,1), first excited (2,2), and second excited (3,3) adiabatic states for the PT-ET model, calculated via MQCL. Middle panel: The percentage of trajectories on each surface as a function of time. Bottom panel: Time evolutions of the (1,2), (2,3), and (1,3) coherences, calculated via MQCL xiii

14 List of Figures 3.8 The percentage of trajectories on each surface as a function of time (top panel) and the time evolutions of the(1,2), (2,3), and(1,3) coherences(bottom panel) for the ET-PT model under the weakest proton/electron-solvent coupling conditions (i.e. C sp = /C se = ) Top panel: Ground (1,1), first excited (2,2), and second excited (3,3) state adiabatic PESs as a function of the solvent coordinate, Q, for the concerted PCET model. The blue curves correspond to these surfaces, the red curves correspond to the mean surfaces between adjacent surfaces, and the green curve corresponds to the mean surface between the ground and second excited state surfaces. Bottom panel: Absolute values of the nonadiabatic coupling matrix elements, d 12, d 23, and d 13 as a function of the solvent coordinate, Q Top panel: Time evolutions of the populations of the ground(1,1), first excited (2,2), and second excited(3,3) adiabatic states for the concerted PCET model, calculated via MQCL. Middle panel: The percentage of trajectories on each surface as a function of time. Bottom panel: Time evolutions of the (1,2), (2,3), and (1,3) coherences, calculated via MQCL The rate coefficient, k AB (t), as a function of time. The dotted line indicates the value of the rate constant, k AB Upper panel: The ground and first-excited state PESs as a function of q s for Model 1. Lower panel: The absolute value of the nonadiabatic coupling between these two surfaces, d 12, as a function of q s The adiabatic (left) and nonadiabatic (right) rate coefficients as a function of time, k AB (t). The dotted red lines indicate the plateau values xiv

15 List of Tables 1.1 Some useful unit conversions Parameters values (in atomic units) for the concerted PCET (top), sequential PT-ET (middle), and ET-PT (bottom) reactions Seven sets of proton/electron-solvent coupling constants (i.e., {C sp,c se }) considered in this study for the three PCET mechanisms, ranging from low (1) to high (7) coupling. All values are in atomic units Three sets of proton/electron-solvent coupling constants (i.e., {C sp,c se }) for the ET-PT mechanism. All values are in atomic units Parameters for the model Hamiltonian in Eq All values are in atomic units except the temperature, T, which is in Kelvin xv

16 Chapter 1 Introduction 1.1 PCET in biology and chemistry Proton-coupled electron transfer (PCET), which is a proton transfer coupled to an electron transfer, is at the heart of important biological processes such as photosynthesis, respiration, and DNA repair [3]. To illustrate the role of PCET, let us consider photosynthesis as an example. Photosynthesis is the process by which the solar energy is converted into the chemical energy stored in chemical bonds of carbohydrates. This solar energy is used to split off water s hydrogen from oxygen. Then, hydrogen is combined with carbon dioxide (absorbed from air or water) to form glucose and release oxygen: 6CO 2 +6H 2 O C 6 H 12 O 6 +6O 2 (1.1) A deep understanding of this ubiquitous reaction may lead to a clean and sustainable solution to the increasing worldwide demand for energy. Thus, we briefly discuss this reaction in more detail. Photosynthesis begins when light is absorbed by proteins called photosynthetic reaction centers found inside chloroplasts (in green plants). The photosynthetic complex, Photosystem II, is the first protein complex involved in solar energy conversion. It is comprised of at least 99 cofactors: 35 chlorophyll a, 12 beta-carotene, two pheophytin, three plastoquinone, two heme, bicarbonate, 25 lipids, and seven n-dodecyl-beta-d-maltoside detergent molecules, 1

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19 Chapter 1. Introduction PCET mechanisms, where either ET precedes PT (i.e., ET-PT) or PT precedes ET (i.e, PT-ET). Among these steps, normally EPT is the one that follows reaction pathways that avoid highenergy intermediates. One should know that EPT is different from HAT, though in some cases this difference can be very subtle. In EPT, electrons and protons transfer from different orbitals on the donor to different orbitals on the acceptor. An example of EPT is illustrated in step 2 of the comproportionation reaction in Fig In this step, a concerted transfer occurs with the electron transferring from a metal-based dπ orbital at Ru II OH 2+ 2 to a dπ acceptor orbital at Ru IV O 2+. Proton transfer occurs from a σ(o H) orbital at Ru(II) to an O-based lone-pair orbital at Ru IV O 2+ [10, 11]. Figure 1.3: An example of a concerted electron-proton transfer (EPT), reprinted (adapted) with permission from Ref. 3. Copyright (2015) American Chemical Society. In EPT, the e /H + donor orbitals and e /H + acceptor orbitals must interact electronically, enabling simultaneous transfer. Normally formation of a hydrogen-bond is the first step for making such an interaction possible, e.g. step 1 in Fig At last, the simultaneous transfer of an electron and proton occurs if this motion is faster than the periods of the coupled vibrations (tens of femtoseconds) and solvent modes ( 1 ps) [12]. On the other hand, in HAT, both the transferring electron and proton come from the same orbital in one of the reactants. Figure 1.4 represents an example of HAT in the oxidation of benzaldeyde by Ru IV (bpy) 2 (py)(o) 2+. The transferring electron and proton both come from the same σ(c H) bond of the benzaldehyde. MS-EPT refers to either transferring of an electron and proton from a donor species to different acceptors or from different donors to one acceptor species. As an example, one 4

20 Chapter 1. Introduction Figure 1.4: An example of hydrogen atom transfer, reprinted (adapted) with permission from Ref. 3. Copyright (2015) American Chemical Society. can refer to the quenching of triplet C 60 and tetracene fluorescence by phenols which is strongly enhanced through addition of pyridines [13]. Based on kinetic measurements and independent spectroscopic data, the quenching is the result of a trimolecular transition state in which an electron transfer occurs from the phenol to the excited molecule while a proton transfers from the phenolic OH to the pyridine nitrogen. It should be noted that the nomenclature used to describe concerted electron-proton transfer, i.e. EPT, has not been standardized and other terms are being used in the literature to describe the same elementary step, including concerted proton-electron transfer (CPET) [14], electron transfer proton transfer(et-pt)[15], concerted electron-proton transfer(cep) [16], and concerted proton-coupled electron transfer (concerted PCET) [17]. The last term is what we use in this thesis to differentiate the concerted mechanism of the transfer of a proton and electron from the sequential mechanisms. A schematic representation of the reactions studied in this thesis are depicted in Fig Sequential PCET ET D H A D + H A PT Concerted PCET PT D + ET H A D H A Sequential PCET Figure 1.5: A schematic representation of concerted and sequential PCET reactions. D represents the donor and A represents the acceptor species. 5

21 Chapter 1. Introduction 1.3 Theoretical challenges Due to the importance of PCET in chemical and biological reactions, it is not surprising that this phenomenon has been the subject of numerous experimental[3, 14, 18 23] and theoretical [18, 24 30] studies over the past two decades. A deep understanding of the mechanisms of PCET reactions, however, is complicated by both the quantum mechanical nature of the transferring protons and electrons and the size and complexity of the environments in which they occur. Ideally, a full quantum dynamical treatment should be used to study these reactions, but the large number of degrees of freedom in the surrounding environment renders such a treatment impossible. To circumvent this problem, simplified models of complex systems may be studied to gain qualitative insights into the PCET mechanisms and/or mixed quantum-classical schemes, which treat the transferring protons and electrons quantum mechanically and the environmental degrees of freedom classically, may be used to gain quantitative information about PCET reaction rates. Another theoretical challenge in the study of PCET reactions is related to the different timescales of motions encountered in these systems. There are three distinct timescales in PCET. Electrons move on the fastest timescale, protons move on an intermediate timescale, and the solvent and heavy atoms of the solute move on the slowest timescale[31]. Hence, it is convenient to treat the transferring electrons and protons quantum mechanically, to include effects such as zero point energy and tunnelling, while the heavy particles may be treated classically. If a distinct separation of timescales exists, then one could invoke two adiabatic approximations for PCET, namely that the electrons respond instantaneously to the proton and solvent motion and, likewise, the protons respond instantaneously to the solvent motion. However, this is not always a valid approximation for PCET reactions since electron transfer reactions in solution are often nonadiabatic with respect to the solvent motion [32 34]. When the electrons do not respond instantaneously to the solvent motion then most likely they will not respond instantaneously to the proton motion either, which is typically faster than the solvent motion. A similar behaviour is expected for proton motion compared to the solvent motion. This is due to the presence of long hydrogen bonds, which results in weak couplings between electron donors and acceptors. Thus, an accurate mixed quantum-classical description of PCET reactions should take all the nonadiabatic couplings in the system into account. 6

22 Chapter 1. Introduction 1.4 Mixed quantum-classical dynamics (MQCD) Mixed quantum-classical dynamics (MQCD) approaches treat a few degrees of freedom (DOF) in a subsystem of interest quantum mechanically, while the remaining DOFs in the environment are treated classically. The last few decades have witnessed the emergence of various mixed quantum-classical approaches for simulating nonadiabatic dynamics of chemical and biological systems. The most well-known of these methods is Fewest Switches Surface Hopping (FSSH), introduced by Tully and Preston in 1971 [35, 36] and later improved in the early 90s [37, 38]. In this approach, an ensemble of classical trajectories is evolved on single adiabatic potential energy surfaces with stochastic nonadiabatic transitions between surfaces. Over the years, improved versions of FSSH have been introduced, including secondquantized surface hopping (SQUASH) [39] and global flux surface hopping (GFSH) [40, 41] to name a few. Other MQCD approaches include mixed quantum-classical Liouville (MQCL) dynamics [56, 58] and mixed quantum-classical Bohmian dynamics [59 61]. Semi-classical approaches have also been introduced, which treat the subsystem and environmental DOF on an equal footing. Such approximate schemes include the classical Wigner (CW) approximation [42 44], centroid molecular dynamics [45], and ring-polymer molecular dynamics (RPMD) [46]. Quantum mechanics/molecular mechanics (QM/MM) [47 51] has also proved useful in simulating large and complex systems. It provides a feasible way for addressing the electronic structure problem, while the aforementioned methods address the molecular dynamics problem Fewest switches surface hopping (FSSH) Tully s FSSH is one of the most widely used mixed quantum-classical methods for simulating the nonadiabatic dynamics of quantum processes in condensed phase systems. Compared to other MQCD methods, it is more computationally efficient, making simulations of large systems feasible. It is also conceptually simple, representing a quantum wave packet in terms of an ensemble of independent trajectories. The multiple steps in an FSSH simulation can be summarized as follows: 7

23 Chapter 1. Introduction 1. The time-independent Schrödinger equation, ĥ α;q = E α(q) α;q, for the quantum subsystem is solved for a fixed configuration of the classical DOF, Q, to obtain the adiabatic energies, E α (Q), and the corresponding adiabatic states, α;q. 2. A time-dependent wave function is defined as Φ(t) = k C k(t) α;q, where C k (t) are complex-valued expansion coefficients. 3. Substitute this wavefunction into the time-dependent Schrödinger equation to obtain i C k = j C j (V kj i Q d kj ) (1.5) where the coupling potential between two states is given by V kj = α k ;Q ĥ(q) α j;q (1.6) and the nonadiabatic coupling vector between the two states is defined as d kj (Q) = α k ;Q Q α j ;Q (1.7) Equation 1.5 is integrated numerically as the trajectory of the classical DOF is propagated on the single state k in order to obtain the amplitudes C k of each quantum mechanical state at a given time t. 4. Hellmann-Feynman forces, the forces on each classical DOF due to the quantum subsystem, are calculated as follows F αk (Q) = α k ;Q ĥ Q α k;q (1.8) These forces along with the remaining classical forces are used to update the positions and velocities of the classical DOF via Newton s equations of motion. 5. The fewest switches algorithm apportions trajectories among states according to the quantum probabilities C k (t) 2 with the minimum required number of quantum transitions. According to this algorithm, the probability of switching from the current state k to all other 8

24 Chapter 1. Introduction states j during the time interval between t and t+ is g kj = b jk(t+ ) a kk (t+ ) (1.9) where a kk = C k Ck is the population of state k and b jk = 2 1 Im(αjk V jk) 2Re(αjk Q d kj ). Whether or not a switch to any state j occurs is determined by comparing g kj to a randomly generated number between 0 and 1. The transition occurs, or is accepted, if the probability of transition is greater than the random number. This switching procedure ensures that, for a large ensemble of trajectories, the fraction of trajectories assigned to any state k at any time t will equal the quantum probability C k (t) If the transition is accepted, then the velocities of the classical DOFs are shifted to conserve energy. If the transition is rejected, then the classical DOFs continue to evolve on the adiabatic PESs corresponding to the current state. 7. Steps 1 to 6 are repeated for each time step of the MD simulation. FSSH has been widely used to study the dynamics and rates of charge transfer reactions including PCET [28, 62 65]; however, there have been some issues with the accuracy of the method [66 69]. To represent a wave packet, one needs to satisfy conservation of the total energy of the ensemble of the trajectories, while allowing energy exchange between individual trajectories. Although each trajectory conserves energy in FSSH, the decoupled classical trajectories do not exchange energy and hence cannot properly represent a quantummechanical wave packet. As a result, FSSH suffers from neglecting decoherence which is the collapse of the quantum superposition when coupled to a classical environment. FSSH evolves the wave function coherently and although there were attempts to add corrections to account for decoherence, they were ad hoc in nature [70, 71]. It has been shown that in the case of the spin-boson model, FSSH yields time-reversible dynamics only with zero friction and for certain initial energies. However, under other conditions, the value of the electronic wave function is not periodic and FSSH is unable to recover the correct Marcus golden rule scaling unless decoherence added to the algorithm [72, 73]. 9

25 Chapter 1. Introduction Mixed quantum-classical Liouville (MQCL) approach The mixed quantum-classical Liouville (MQCL) surface-hopping algorithm is based on the numerical solution of the quantum-classical Liouville equation (QCLE) [74 85], which rigorously accounts for quantum coherence/decoherence effects in mixed quantum-classical dynamics. In the MQCL approach, the classical DOF are evolved either on single adiabatic potential energy surfaces or on the mean of two adiabatic surfaces corresponding to two coherently coupled states [83, 84], as opposed to just on single adiabatic surfaces as in FSSH. To illustrate the differences between these two approaches for calculating condensed phase charge transfer rates, it has been found [87] that the MQCL approach yields a proton transfer rate constant for the Azzouz-Borgis model [86] of a proton transfer reaction in a phenol trimethylamine complex dissolved in CH 3 Cl that is approximately twice as large as that given by FSSH [88]. Owing to its more rigorous treatment of coherence/decoherence effects, the MQCL approach is expected to yield more accurate rate constants for condensed phase PCET reactions and may also shed new light on their underlying mechanisms. As in all MQCD approaches, one considers a quantum subsystem that is coupled to a classical environment, whose Hamiltonian is given by Ĥ(Q,P) = P2 2M +V e(q)+ ˆp2 2m + ˆV s (ˆq)+ ˆV c (ˆq,Q) H e (Q,P)+ĥs(ˆq, ˆp)+ ˆV c (ˆq,Q) H e (Q,P)+ĥ(Q), (1.10) where m, ˆq, and ˆp are the vectors of masses, positions, and momenta of the n quantum DOF, respectively; M, Q, and P are the vectors of masses, positions, and momenta of the N classical DOF, respectively; H e = P2 2M + V e and ĥs = ˆp2 2m + ˆV s are the environment and subsystem Hamiltonians, respectively; ˆVc is the subystem-environment coupling potential energy and ĥ = ĥs + ˆV c [throughout this text, operators are capped (e.g., Â)]. The state of this mixed quantum-classical system can be represented in terms of the partial Wigner transform [89] of the density operator, ˆρ, over the environmental DOF: ˆρ W (X,t) = ( ) 1 N 2π dze ip Z/ Q Z/2 ˆρ(t) Q+Z/2, (1.11) 10

26 Chapter 1. Introduction where X = (Q,P). When m/m 1, the dynamics of ˆρ W (X,t) can be accurately described by the QCLE [58, 79]: tˆρ W(X,t) = i [ĤW (X,t), ˆρ W (X,t)] + 1 } ({ĤW (X,t), ˆρ W (X,t) 2 { }) ˆρ W (X,t),ĤW (X,t), (1.12) where [ ] is the commutator and the Poisson bracket { } is defined as {ÂW (X,t), ˆB P W (X,t)} = (X,t)( ÂW Q ) Q P ˆBW (X,t). (1.13) Here, Q/P and Q/P are gradient operators with respect to Q/P which act on the term to the right and left of them, respectively. The quantum subsystem may be represented in terms of an adiabatic basis, α; Q, which are the solutions of ĥ(q) α;q = E α(q) α;q. In this representation, the QCLE is given by [79] ρ αα W (X,t) t = i L αα,ββ ρββ W (X,t), (1.14) ββ where the evolution operator is defined as il αα,ββ (iω αα +il αα )δ αβδ α β J αα,ββ. (1.15) In the above equation, the classical Liouville operator, il αα, is given by il αα = P M Q + 1 ( 2 F α W +F α W ) P. (1.16) When the quantum state indices are equal (i.e., α = α ), the classical evolution is carried out subject to the Hellmann-Feynman forces, FW α = α;q ˆV c(ˆq,q) Q α;q, on a single adiabatic surface E α (Q). However, when the quantum state indices are different (i.e., α α ), the classical evolution is carried out on the mean of two adiabatic surfaces, [E α (Q)+E α(q)]/2, accompanied by quantum phase oscillations of frequency ω αα = (E α E α )/. The term responsible for nonadiabatic transitions and the associated changes in the bath momentum 11

27 Chapter 1. Introduction (to conserve energy) is given by J αα,ββ (t) = P M d αβ ( S αβ ) P P M d α β ( S α β P δ α β ) δ αβ, (1.17) where S αβ = FW αδ αβ F αβ W ( P M d αβ) 1 = E αβ d αβ ( P M d αβ) 1, d αβ = α;q Q β;q is the nonadiabatic coupling vector matrix element, and F αβ W is the off-diagonal matrix element of the force. Formally, the solution of the QCLE for ρw αα (X,t) can be written as ρ αα W (X,t) = ( ) e iˆlt ββ αα,ββ ρββ (X). (1.18) W A numerical solution may be obtained using the sequential short-time propagation (SSTP) algorithm[83], which is based on a decomposition of the propagator into short-time segments. Dividing the time interval t into S segments such that the j th segment has length t j = t j t j 1 = t (where t is the time step), gives ρ α 0α 0 W (X,t) = (α 1 α 1 )...(α Sα S ) [ S ] (e iˆl t j ) αj 1 α j 1,α jα j j=1 where the short-time propagator is approximated by ρ α Sα S W (X), (1.19) (e iˆl t j ) αj 1 α j 1,α jα j W αj 1 α (t j 1,t j 1 j )e il α j 1 α t j j 1 ( ) δ αj 1 α j δ α j 1 α + tj j α j 1 α j 1,α jα j (1.20) and W αj 1 α j 1 (t j 1,t j ) = e iω α j 1 α j 1 t j is the phase factor for that segment. In the SSTP algorithm, Monte Carlo sampling is used to evaluate the multi-dimensional sums over quantum indices in Equation A typical surface-hopping trajectory generated by this algorithm is composed of segments in which the classical DOF evolve on a single adiabatic surface (when α j = α j ) and on the mean of two surfaces (when α j α j ), as governed by the first term in Equation These segments are interrupted by nonadiabatic transitions, as governed by the second term in Equation 1.20, which cause the classical DOF to hop to a new surface (or mean surface), followed by evolution on this surface. The details for 12

28 Chapter 1. Introduction propagating a system starting from an initial condition {X,α 0,α 0 } via the SSTP algorithm are provided in Refs. 83 and FSSH vs. MQCL In this section, we point out the advantages of the MQCL approach over FSSH: 1) FSSH was not derived from the quantum Liouville equation, but instead has been justified on empirical grounds. On the other hand, the MQCL equation was rigorously derived from the quantum Liouville equation in the limit of m M quantum and classical DOFs, respectively. << 1, where m and M are the masses of 2) The time evolution of FSSH classical trajectories occurs on single adiabatic potential energy surfaces (PESs), while in MQCL, the evolution occurs not only on single PESs but also on the average of these PESs. 3) FSSH is incapable of treating decoherence, which as mentioned previously stems from the lack of energy exchange between independent trajectories. In MQCL, while a trajectory evolves on a mean surface the observable in question acquires a phase factor [83, 84]. Fluctuations of the classical coordinates can cause this phase factor to oscillate in time differently for each trajectory, so averaging over an ensemble of trajectories can lead to destructive interference in the expectation value of an observable. In this sense, MQCL is able to treat decoherence. 4) FSSH is also incapable of treating recoherence properly while owing to the mean surface evolution, MQCL is capable of capturing recoherence. 1.5 Outline of thesis Motivated by the need to properly incorporate all nonadiabatic couplings involved in a PCET reaction and to properly treat coherence/decoherence effects, our method of choice in this thesis ismqcldynamics. InChapter 2, weapply MQCLdynamicsto thestudy ofareduced model of a PCET reaction. Since this work constitutes the first application of the MQCL approach to PCET, the model is chosen to be the same as that in Refs. 90, 91, and 92 13

29 Chapter 2. PCET via MQCL (which is based on a charge transfer model developed in Ref. 93). By varying the parameters in the model Hamiltonian, PCET can either occur concertedly or sequentially between two positively charged ions. In Chapter 3, we use MQCL dynamics to calculate the adiabatic state populations for the same model. The results in this chapter will show a novel picture of nonadiabatic PCET reactions, thanks to MQCL dynamics. In Chapter 4, we derive a general, MQCL-based expression for the time-dependent rate coefficient of a PCET reaction and apply it to an improved version of the previously studied model to calculate the rate constant of a concerted PCET reaction. This rate constant is compared to the exact one for this model to assess the validity of our expression. Conclusions and future work are described in Chapter 5. Since atomic units are used commonly in this thesis, Table I contains several useful conversions from atomic units to SI and common units. Table 1.1: Some useful unit conversions. One atomic unit (a.u.) SI units Common units Length m 5.29 Å Energy J kcal/mol Time J fs Velocity m/s 14

30 Chapter 2 An Analysis of Model Proton-Coupled Electron Transfer Reactions via the Mixed Quantum-Classical Liouville Approach 2.1 Introduction A deep understanding of the mechanisms of PCET reactions is complicated by both the quantum mechanical nature of the transferring protons and electrons and the size and complexity of the environments in which they occur. To circumvent this problem, simplified models of complex systems may be studied to gain qualitative insights into the PCET mechanisms and/or mixed quantum-classical schemes may be used to gain quantitative information about PCET reaction rates. Tully s FSSH [37] is one of the most widely used mixed quantum-classical methods for simulating the nonadiabatic dynamics of quantum processes in condensed phase systems. This method was applied to PCET for the first time in 1997 by Fang and Hammes-Schiffer [90]. 15

31 Chapter 2. PCET via MQCL They studied a simple model containing three coupled degrees of freedom, which represent a proton, an electron, and a solvent mode. The proton and electron were treated quantum mechanically, while the solvent mode was treated classically. By varying the model parameters, the authors investigated the nonadiabatic dynamics of both sequential (viz., ET followed by PT and PT followed by ET) and concerted PCET mechanisms. In a subsequent study [91], several different methods for incorporating decoherence into FSSH (which, on its own, evolves the wave function coherently) were applied to a similar model exhibiting a large number of avoided curve crossings. It was found that the agreement between the FSSH and exact quantum results for the adiabatic state populations was excellent at very short times (up to 19 fs), but at longer times, the results begin to differ and remain significantly different for the duration of the time (i.e., 168 fs). This deviation is most likely due to an improper treatment of decoherence. In this Chapter, we employ for the first time a surface-hopping algorithm based on the numerical solution of the quantum-classical Liouville equation which is called MQCL [74 85] to study PCET. Since the MQCL inherently and rigorously accounts for quantum coherence/decoherence effects in mixed quantum-classical systems, it is expected to yield more accurate rate constants for condensed phase PCET reactions and may also shed new light on their underlying mechanisms. In fact, for the Azzouz-Borgis model of a proton transfer reaction in a phenol trimethylamine complex dissolved in CH 3 Cl [86], it was found that the MQCL gives a proton transfer rate constant that is approximately twice as large [87] as that given by FSSH [88], thereby highlighting the differences between the two approaches for treating condensed phase charge transfer reactions. Since this work constitutes the first application of the MQCL approach to PCET, we chose to study the same model as in Refs. 90, 91, and 92 (based on a charge transfer model developed in Ref. 93) to (1) demonstrate the applicability of this method to PCET reactions, (2) compare and contrast, wherever possible, with the results of the standard FSSH approach in Ref. 90, and (3) investigate the role played by mean surface evolution in the concerted and sequential mechanisms for a range of subsystem-bath couplings. This chapter is organized as follows: The three PCET models studied in this work are presented in Section 2. Section 3 summarizes the simulation details for generating nonadiabatic quantum-classical dynamics. 16

32 Chapter 2. PCET via MQCL Section 4 presents and discusses the results for the model PCET reactions and concluding remarks are made in Section PCET model The reduced model of a PCET reaction considered in this work contains three coupled degrees of freedom, representing a proton, an electron, and a solvent mode. The PCET reaction occurs between two positively charged ions, referred to as the donor (D) and acceptor (A), which are separated by a fixed distance d DA. The coordinates of the proton and electron, denoted by q p and q e, respectively, are measured relative to D and they move from D to A only in one dimension, along the D A axis (see Fig. 2.1). Figure 2.1: Schematic representation of the one-dimensional PCET model studied in this work. The protonic and electronic potentials (V p and V e, respectively) are superimposed onto the donor(d) acceptor(a) axis. The protonic and electronic coordinates (q p and q e, respectively) are defined relative to the donor. The coordinate of the collective solvent mode to which the proton and electron are coupled, is denoted by Q. Within our approach, the proton and electron are treated quantum mechanically, whereas the solvent mode is treated classically. The Hamiltonian of this mixed quantum-classical system is given by Ĥ = ˆK p + ˆK e +K s + ˆV p (q p )+ ˆV e (q e )+ ˆV pe (q p,q e )+ ˆV pes (q p,q e,q)+v s (Q) ĥ+k s. (2.1) Here, ˆK p and ˆK e denote the kinetic energy operators of the proton and electron, respectively, K s is the kinetic energy of the solvent mode, and ĥ = Ĥ K s. The protonic potential energy 17

33 Chapter 2. PCET via MQCL operator ˆV p, as can be inferred from Fig. 2.1, is a double-well potential given by ˆV p (q p ) = [ 12 E q 4 p (a 2 a 1 ) 3 (2a 3 a 1 a 2 ) 4 (a 1 +a 2 +a 3 ) q3 p 3 + (a 1 a 2 +a 1 a 3 +a 2 a 3 ) q2 p 2 (a 1a 2 a 3 )q p + a2 2 (a2 2 2a 2(a 1 +a 3 )+6a 1 a 3 ) 12 ], (2.2) where a 1 and a 3 correspond to the positions of the two minima, where proton is near the donor or acceptor, respectively. a 2 corresponds to the position of the maximum between a 1 and a 3 and E is the energy difference between the minimum at a 1 and a 2. The electronic potential energy operator, ˆVe, corresponding to the Coulombic interactions between the electron and the donor and acceptor charges, is given by ˆV e (q e ) = C ec D erf(q ed /ξ ed ) q ed C ec A erf(q ea /ξ ea ) q ea +e 5.0(qe+5.0) +e 5.0(d DA q e+5.0), (2.3) where C e, C D, and C A are the partial charges of the electron, donor, and acceptor, respectively, and q ed and q ea are the distances between the electron and the donor and acceptor, respectively. In this work, both ξ ed and ξ ea are chosen to be equal to 1. The two last terms are added for numerical stability, they are repulsive terms that prevent the electron from travelling too far beyond the donor and acceptor. The potential energy operator corresponding to the Coulombic interaction between the proton and electron is given by ˆV pe (q p,q e ) = C pc e erf(q pe /ξ pe ) q pe, (2.4) where q pe = q p q e is the distance between the proton and electron. The coupling between the solvent coordinate and the proton and electron is bilinear in form and is given by the following operator: ˆV pes (q p,q e,q) = C sp (Q Q o p)(q p q o p) C se (Q Q o e)(q e q o e), (2.5) where C sp and C se are the coupling parameters of the proton and electron to the solvent, respectively, and Q o p, Q o e, q o p, and q o e are free parameters. When Q < Q o p/e and q < qo p/e, the proton/electron is stabilized near the donor. Conversely, if Q > Q o p/e and q > qo p/e, the 18

34 Chapter 2. PCET via MQCL proton/electron is stabilized near the acceptor. Finally, the dynamics of the classical solvent mode is governed by a harmonic potential of the form V s (Q) = 1 2 m sω s 2 (Q Q o ) 2, (2.6) where m s and ω s are the mass and frequency, respectively, of the collective solvent mode. It should be noted that for Hamiltonians with bath terms that are harmonic and subsystembath coupling terms that are linear in the bath coordinates, MQCL dynamics is exact. The only approximation made in the sequential short-time propagation (SSTP) solution of the MQCL equation is the momentum-jump approximation, which has been shown numerous times to be a very good approximation. By varying the parameters in the Hamiltonian, one can generate systems in which the proton and electron transfer take place in a concerted fashion and systems in which the transfer is sequential, i.e., PT before ET (i.e., PT-ET) and vice versa (i.e., ET-PT). The parameter values used to investigate the concerted, sequential PT-ET, and sequential ET-PT mechanisms are presented in Table 1. The main differences between the three sets of parameter values are as follows: The partial charges on the proton and electron are chosen to be higher (i.e., ±0.32) in the concerted case than in the sequential cases (i.e., ±0.15) since the attraction between the proton and electron should be higher for them to transfer together. In order for the proton and electron to transfer in a concerted fashion, they should also be similarly coupled to the solvent and therefore their respective couplings are chosen to be equal (i.e., C sp = C se ). In contrast, in the case of sequential transfer, the proton and electron should experience different degrees of coupling to the solvent. In addition to the differences in the couplings to the solvent, the parameters Q o p, Q o e, and Q o are different for each mechanism. Namely, Q o p = Q o e = Q o in the case of concerted PCET, Q o p < Q o and Q o e > Q o for PT-ET, and Q o e < Q o and Q o p > Q o for ET-PT. In all cases, the midpoint of the donor acceptor distance coincides with the barrier tops of the protonic and electronic potentials (i.e., qp o = qe o = d DA /2). The validity of these parameters is confirmed by examining the 2D potential energy surfaces (PESs) as a function of q p and q e in Fig For example, if one examines the 2D PES of concerted PCET, top row in Fig. 2.2, at Q = 0.4 a.u., which is before the midpoint of the D A distance or the barrier tops of the protonic and electronic potentials, the well at 19

35 Chapter 2. PCET via MQCL Table 2.1: Parameters values (in atomic units) for the concerted PCET (top), sequential PT-ET (middle), and ET-PT (bottom) reactions. m s = Q o = 0.0 ω s = a 1 = 2.5 a 2 = 3.0 a 3 = 3.5 E = d DA = 6.0 C D = 0.6 C A = 0.6 C p = 0.32 C e = 0.32 qp o = 3.0 qe o = 3.0 Q o p = 0.0 Q o e = 0.0 C sp = C se = m s = Q o = 0.4 ω s = a 1 = 3.5 a 2 = 4.0 a 3 = 4.55 E = d DA = 8.0 C D = 0.6 C A = 0.6 C p = 0.15 C e = 0.15 qp o = 4.0 qe o = 4.0 Q o p = 0.6 Q o e = 0.0 C sp = C se = m s = Q o = 0.3 ω s = a 1 = 3.5 a 2 = 4.0 a 3 = 4.5 E = d DA = 8.0 C D = 0.55 C A = 0.55 C p = 0.15 C e = 0.15 qp o = 4.0 qe o = 4.0 Q o p = 0.0 Q o e = 0.6 C sp = C se = q p = 2.5 a.u./q e = 0.0 a.u. has the lowest depth. It means that both proton and electron are near the donor. On the other hand, at Q = +0.4 a.u., which is after the barrier tops of the protonic and electronic potentials, the well at q p = 3.5 a.u./q e = 6.0 a.u. has the lowest depth. It means that both proton and electron are near the acceptor. At Q = 0.0 a.u., which coincides with the barrier tops of the potentials, both of these wells show the same depth, i.e. both proton and electron are at the midpoint between the donor and the acceptor. 2.3 Simulation details To carry out the SSTP algorithm, one needs to solve the time-independent Schrödinger equation at each step of the simulation: ĥ α;q = E α (Q) α;q, (2.7) where ĥ is the model Hamiltonian given in Eq This involves expanding the adiabatic wave function α; Q in an orthonormal set of two-particle basis functions as α;q = m,nc α mn φ p(m) φ e(n), (2.8) 20

36 Chapter 2. PCET via MQCL Q = Q = 0.0 Q = Concerted q p Q = Q = Q = PT-ET q p Q = Q = Q = ET-PT q p q e q e q e Figure 2.2: The 2D PESs as a function of q p and q e at different solvent configurations for concerted PCET (top row), sequential PT-ET (middle row), and the sequential ET-PT (bottom row) mechanisms. Q values are in Å but q p and q e values in this figure are given in a.u. where φ p(m) and φ e(n) are one-particle basis functions and are chosen to be the solutions of the quantum harmonic oscillator for a proton and an electron, respectively, i.e., φ p(m) (q p ) = q p φ p(m) = (2 m m! π) 1/2 α 1/2 p e α2 p (qp qo p )/2 H m (α p (q p q o p)), (2.9) and φ e(n) (q e ) = q e φ e(n) = (2 n n! π) 1/2 α 1/2 e e α2 e (qe qo e )/2 H n (α e (q e q o e)), (2.10) 21